Symplectic singularities, color confinement, and the quantum Dirac sheaf

A bstract A singularity ℂ 2 r / G , with G a split symplectic reflection group, may or may not be crepant. Then the total space 𝒳 of the Donagi-Witten integrable system is crepant for some 4d N=2 N = 2 SCFT and non-crepant for others. Which physical mechanism controls the (dis)crepancy? Surprisingly, it is the detailed physics of color confinement (and its generalizations for non-Lagrangian QFT). A 4d N=2 N = 2 SCFT carries a Frobenius algebra ℛ, the quantum cohomology ring of 𝒳 (defined via mirror symmetry), and 𝒳 is crepant iff its central Witten index dim ℛ is equal to its Euler number χ (𝒳). When the SCFT has a Lagrangian, ℛ is fixed by compatibility with confinement, and physics may require a discrepancy to be present. The quantum cohomology depends on quantum-geometric data, and a classical Seiberg-Witten geometry may have several inequivalent ℛ: a relevant quantum datum is the Dirac sheaf ℒ which refines Dirac charge quantization. We get several other results of independent interest, and we fully classify all special geometries of ⋆-type in rank r > 6.